Alfvén-wave transmission through super-Alfvénic parallel relativistic shock waves is studied. We calculate the wave transmission coefficients for given shock properties. We show (i) that the Alfvén waves downstream the shock wave are propagating predominantly anti-parallel to the flow direction for low-Mach-number shocks, as in the case of non-relativistic shocks; and (ii) that for high-Mach-number ultra-relativistic shocks the forward and backward downstream waves are in equipartition. For low Alfvénic Mach numbers, the scattering center compression ratio of the shock, thus, becomes large and the spectral index of accelerated test particles approaches the limit σ → 1 at shock waves approaching the critical value of the quasi-Newtonian Alfvénic Mach number (i.e., the ratio of upstream fluid and Alfvén proper speeds), which depends on the shock properties, and equals the square root of the compression ratio at the test-wave limit. Although the inclusion of the wave electromagnetic and velocity fields to the shock jump conditions is likely to decrease the scattering-center compression ratio for shocks with critical Mach numbers, values significantly above the gas compression ratio can be expected for such shocks. Particle acceleration in weak relativistic shocks propagating in magnetized astrophysical jets may, therefore, be substantially more efficient than predicted by models neglecting turbulent electric fields.